||The design of caching algorithms to maximize hit probability has been extensively studied. In this paper, we associate each content with a utility, which is a function of either the corresponding content hit rate or hit probability. We formulate a cache optimization problem to maximize the sum of utilities over all contents under stationary and ergodic request processes. This problem is non-convex in general but we reformulate it as a convex optimization problem when the inter-request time (irt) distribution has a non-increasing hazard rate function. We provide explicit optimal solutions for some irt distributions, and compare the solutions of the hit-rate based (HRB) and hit-probability based (HPB) problems. We formulate a reverse engineering based dual implementation of LRU under stationary arrivals. We also propose decentralized algorithms that can be implemented using limited information and use a discrete time Lyapunov technique (DTLT) to correctly characterize their stability. We find that decentralized algorithms that solve HRB are more robust than decentralized HPB algorithms. Informed by these results, we further propose lightweight Poisson approximate decentralized and online algorithms that are accurate and efficient in achieving optimal hit rates and hit probabilities.